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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity.
Our main technical contribution is a novel balanced separator theorem for planar δ-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed δ ⩾ 0, we can find a small balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph.
An important advantage of our separator is that the union of our separator (vertex set Z) with any subset of the connected components of G - Z induces again a planar δ-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that the size of the separator is poly(δ) ⋅ log n.
As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar δ-hyperbolic graphs. We prove that both Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant δ, running in n polylog(n) ⋅ 2^𝒪(δ²) ⋅ ε^{-𝒪(δ)} time.
We also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no n^{o(δ)}-time algorithm on planar δ-hyperbolic graphs, unless ETH fails.

Sándor Kisfaludi-Bak, Jana Masaříková, Erik Jan van Leeuwen, Bartosz Walczak, and Karol Węgrzycki. Separator Theorem and Algorithms for Planar Hyperbolic Graphs. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 67:1-67:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kisfaludibak_et_al:LIPIcs.SoCG.2024.67, author = {Kisfaludi-Bak, S\'{a}ndor and Masa\v{r}{\'\i}kov\'{a}, Jana and van Leeuwen, Erik Jan and Walczak, Bartosz and W\k{e}grzycki, Karol}, title = {{Separator Theorem and Algorithms for Planar Hyperbolic Graphs}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {67:1--67:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.67}, URN = {urn:nbn:de:0030-drops-200126}, doi = {10.4230/LIPIcs.SoCG.2024.67}, annote = {Keywords: Hyperbolic metric, Planar Graphs, r-Division, Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

For smooth convex disks A, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes G^{hom}(A) and G^{sim}(A) of intersection graphs that can be obtained from homothets and similarities of A, respectively. Namely, we prove that G^{hom}(A) = G^{hom}(B) if and only if A and B are affine equivalent, and G^{sim}(A) = G^{sim}(B) if and only if A and B are similar.

Mikkel Abrahamsen and Bartosz Walczak. Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2023.2, author = {Abrahamsen, Mikkel and Walczak, Bartosz}, title = {{Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {2:1--2:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.2}, URN = {urn:nbn:de:0030-drops-178523}, doi = {10.4230/LIPIcs.SoCG.2023.2}, annote = {Keywords: geometric intersection graph, convex disk, homothet, similarity} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel’s circle problem (1959). The proof relies on a (multidimensional) version of Gallai’s theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.

James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak. A Solution to Ringel’s Circle Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{davies_et_al:LIPIcs.SoCG.2022.33, author = {Davies, James and Keller, Chaya and Kleist, Linda and Smorodinsky, Shakhar and Walczak, Bartosz}, title = {{A Solution to Ringel’s Circle Problem}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {33:1--33:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.33}, URN = {urn:nbn:de:0030-drops-160413}, doi = {10.4230/LIPIcs.SoCG.2022.33}, annote = {Keywords: circle arrangement, chromatic number, Gallai’s theorem, polynomial method} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3⋅4^{ω-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time.

James Davies, Tomasz Krawczyk, Rose McCarty, and Bartosz Walczak. Colouring Polygon Visibility Graphs and Their Generalizations. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{davies_et_al:LIPIcs.SoCG.2021.29, author = {Davies, James and Krawczyk, Tomasz and McCarty, Rose and Walczak, Bartosz}, title = {{Colouring Polygon Visibility Graphs and Their Generalizations}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {29:1--29:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.29}, URN = {urn:nbn:de:0030-drops-138281}, doi = {10.4230/LIPIcs.SoCG.2021.29}, annote = {Keywords: Visibility graphs, \chi-boundedness, pseudoline arrangements, ordered graphs} }

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**Published in:** LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Let C and D be hereditary graph classes. Consider the following problem: given a graph G in D, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to C. We prove that it can be solved in 2^{o(n)} time, where n is the number of vertices of G, if the following conditions are satisfied:
- the graphs in C are sparse, i.e., they have linearly many edges in terms of the number of vertices;
- the graphs in D admit balanced separators of size governed by their density, e.g., O(Delta) or O(sqrt{m}), where Delta and m denote the maximum degree and the number of edges, respectively; and
- the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph.
This leads, for example, to the following corollaries for specific classes C and D:
- a largest induced forest in a P_t-free graph can be found in 2^{O~(n^{2/3})} time, for every fixed t; and
- a largest induced planar graph in a string graph can be found in 2^{O~(n^{3/4})} time.

Jana Novotná, Karolina Okrasa, Michał Pilipczuk, Paweł Rzążewski, Erik Jan van Leeuwen, and Bartosz Walczak. Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 23:1-23:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{novotna_et_al:LIPIcs.IPEC.2019.23, author = {Novotn\'{a}, Jana and Okrasa, Karolina and Pilipczuk, Micha{\l} and Rz\k{a}\.{z}ewski, Pawe{\l} and van Leeuwen, Erik Jan and Walczak, Bartosz}, title = {{Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs}}, booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)}, pages = {23:1--23:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-129-0}, ISSN = {1868-8969}, year = {2019}, volume = {148}, editor = {Jansen, Bart M. P. and Telle, Jan Arne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.23}, URN = {urn:nbn:de:0030-drops-114845}, doi = {10.4230/LIPIcs.IPEC.2019.23}, annote = {Keywords: subexponential algorithm, feedback vertex set, P\underlinet-free graphs, string graphs} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

We prove that for every integer t greater than or equal to 1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is chi-bounded. This is essentially the strongest chi-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers k > 1 and t > 0, every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has O(n log n) edges.

Alexandre Rok and Bartosz Walczak. Coloring Curves That Cross a Fixed Curve. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{rok_et_al:LIPIcs.SoCG.2017.56, author = {Rok, Alexandre and Walczak, Bartosz}, title = {{Coloring Curves That Cross a Fixed Curve}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {56:1--56:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.56}, URN = {urn:nbn:de:0030-drops-71788}, doi = {10.4230/LIPIcs.SoCG.2017.56}, annote = {Keywords: String graphs, chi-boundedness, k-quasi-planar graphs} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We describe the first algorithm to compute the outer common tangents of two disjoint simple polygons using linear time and only constant workspace. A tangent of a polygon is a line touching the polygon such that all of the polygon lies on the same side of the line. An outer common tangent of two polygons is a tangent of both polygons such that the polygons lie on the same side of the tangent. Each polygon is given as a read-only array of its corners in cyclic order. The algorithm detects if an outer common tangent does not exist, which is the case if and only if the convex hull of one of the polygons is contained in the convex hull of the other. Otherwise, two corners defining an outer common tangent are returned.

Mikkel Abrahamsen and Bartosz Walczak. Outer Common Tangents and Nesting of Convex Hulls in Linear Time and Constant Workspace. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{abrahamsen_et_al:LIPIcs.ESA.2016.4, author = {Abrahamsen, Mikkel and Walczak, Bartosz}, title = {{Outer Common Tangents and Nesting of Convex Hulls in Linear Time and Constant Workspace}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {4:1--4:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.4}, URN = {urn:nbn:de:0030-drops-63465}, doi = {10.4230/LIPIcs.ESA.2016.4}, annote = {Keywords: simple polygon, common tangent, optimal algorithm, constant workspace} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Let S be a subset of R^d with finite positive Lebesgue measure. The Beer index of convexity b(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio c(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate a relationship between these two natural measures of convexity of S.
We show that every subset S of the plane with simply connected components satisfies b(S) <= alpha c(S) for an absolute constant alpha, provided b(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. asserting that this estimate holds for simple polygons.
We also consider higher-order generalizations of b(S). For 1 <= k <= d, the k-index of convexity b_k(S) of a subset S of R^d is the probability that the convex hull of a (k+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every d >= 2 there is a constant beta(d) > 0 such that every subset S of R^d satisfies b_d(S) <= beta c(S), provided b_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma(d) > 0 such that for every epsilon from (0,1] there is a subset S of R^d of Lebesgue measure one satisfying c(S) <= epsilon and b_d(S) >= (gamma epsilon)/log_2(1/epsilon) >= (gamma c(S))/log_2(1/c(S)).

Martin Balko, Vít Jelínek, Pavel Valtr, and Bartosz Walczak. On the Beer Index of Convexity and Its Variants. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 406-420, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{balko_et_al:LIPIcs.SOCG.2015.406, author = {Balko, Martin and Jel{\'\i}nek, V{\'\i}t and Valtr, Pavel and Walczak, Bartosz}, title = {{On the Beer Index of Convexity and Its Variants}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {406--420}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.406}, URN = {urn:nbn:de:0030-drops-51229}, doi = {10.4230/LIPIcs.SOCG.2015.406}, annote = {Keywords: Beer index of convexity, convexity ratio, convexity measure, visibility} }